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Many recently developed Bayesian methods have focused on sparse signaldetection. However, much less work has been done addressing the naturalfollow-up question: how to make valid inferences for the magnitude of thosesignals after selection. Ordinary Bayesian credible intervals suffer fromselection bias, owing to the fact that the target of inference is chosenadaptively. Existing Bayesian approaches for correcting this bias producecredible intervals with poor frequentist properties, while existing frequentistapproaches require sacrificing the benefits of shrinkage typical in Bayesianmethods, resulting in confidence intervals that are needlessly wide. We addressthis gap by proposing a nonparametric empirical-Bayes approach for constructingoptimal selection-adjusted confidence sets. Our method produces confidence setsthat are as short as possible on average, while both adjusting for selectionand maintaining exact frequentist coverage uniformly over the parameter space.Our main theoretical result establishes an important consistency property ofour procedure: that under mild conditions, it asymptotically converges to theresults of an oracle-Bayes analysis in which the prior distribution of signalsizes is known exactly. Across a series of examples, the method outperformsexisting frequentist techniques for post-selection inference, producingconfidence sets that are notably shorter but with the same coverage guarantee.

Optimal post-selection inference for sparse signals: a nonparametric empirical-Bayes approach